Christopher Schafhauser

Abstract We consider correlations, $p_{n,x}$, arising from measuring a maximally entangled state using $n$ measurements with two outcomes each, constructed from $n$ projections that add up to $xI$. We show that the correlations $p_{n,x}$ robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers-Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an ``approximate" representation of the relevant algebra to an exact one. For n=4, the correlations $p_{n,x}$ self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank. The only other family of constant-size self-tests for strategies of unbounded dimension is due to Fu (QIP 2020) who presents such self-tests for an infinite family of maximally entangled states with \emph{even} local dimension. Therefore, we are the first to exhibit a constant-size self-test for measurements of unbounded dimension as well as all maximally entangled states with odd local dimension. In addition, correlations $p_{4,x}$ represent the first self-tests for measurements of rank higher than one.